In order to prepare a 2020 state return you will need the  CD/download software for 2020---it is not possible to use online for a 2020 return.   If you used TT for your 2020 federal return you can download the 2020 desktop software to a full PC or Mac--not to a mobile device or phone.

https://ttlc.intuit.com/questions/1900990-how-do-i-file-a-prior-year-tax-return

And...the 2020 return will have to be printed and mailed.   It cannot be e-filed.

I just need to file state only for 2020, since I already filed federal 2020 with TT online back in April 2021. Will the software pull my 2020 information and put in on my state? I ask because I do not have my W-2s with me.

Information flows from the federal return to the state return.   You have to use exactly the same account and user ID to access the tax file for the federal and you will have to pay for the state software.   And....when you mail a tax return you are supposed to attach copies of your W-2's and/or any 1099's that show tax withheld.   Try getting copies from your employer.